I built a NAP-140 amplifier which I run from a 25-0-25 transformer. It works fine.

I want to include a preamplifier in the case, so I incorporated a separate 12-0-12 toroidal transformer.

Im having trouble wiring the preamp and the amplifier together. Whenever I do so, the 12V toroid starts humming and gets very hot. I measured the voltage between the two 0V connections and found 7V difference. Obviously the smaller transformer is being forced to deliver current.

Both of the transformers are floating (i.e. the 0V is not connected to the chassis ground).

Hi,
the 12-0-12Vac transformer needs it's own PSU and this PSU powers the preamplifier only.

The existing 25-0-25Vac transformer already has it's PSU and this powers the power amplifier only.

The chassis, if any is metal must be connected to Protective Earth (PE).
The point/bolt/welded connection for making this contact is being called the Safety Earth, to differentiate it from the various other grounds that get used and abused in electronics gear.

All exposed conductive parts must be (electrically) connected to the Safety Earth.
That usually means that the centre tap of your main transformer must be connected to somewhere on the chassis using a connection that can pass Mains Fault Current to PE. This allows the fuse to blow and isolate the damaged gear and thus to be safer to touch (not to probe about inside).

If any part of the pre-amp has any exposed conductive parts then it too must be connected to chassis to blow your mains fuse in event of a mains fault.

If you are unsure about this Safety stuff, then seek expert local help or keep asking questions until it becomes absolutely clear, before you start building mains powered gear.

__________________
regards Andrew T.
Sent from my desktop computer using a keyboard

You can't couple two amplifiers which are floating except via a transformer. This means that the two secondaries will, at least indirectly, be connected somehow. The trick is to connect them in the right way.

Why not show us a diagram of what you did? Then maybe we can tell you what you should do. We are good, but we can't read minds.